Calculus Examples

$\int (x - 1) \sin (π x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x - 1$ and $d v = \sin (π x)$ .

$(x - 1) (- \frac{1}{π} \cos (π x)) - \int - \frac{1}{π} \cos (π x) d x$

Step 2

Since $- \frac{1}{π}$ is constant with respect to $x$ , move $- \frac{1}{π}$ out of the integral.

$(x - 1) (- \frac{1}{π} \cos (π x)) - (- \frac{1}{π} \int \cos (π x) d x)$

Step 3

Simplify.

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Step 3.1

Combine $\cos (π x)$ and $\frac{1}{π}$ .

$(x - 1) (- \frac{\cos (π x)}{π}) - (- \frac{1}{π} \int \cos (π x) d x)$

Step 3.2

Multiply $- 1$ by $- 1$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + 1 (\frac{1}{π} \int \cos (π x) d x)$

Step 3.3

Multiply $\frac{1}{π}$ by $1$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π} \int \cos (π x) d x$

Step 4

Let $u = π x$ . Then $d u = π d x$ , so $\frac{1}{π} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = π x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 4.1.2

Since $π$ is constant with respect to $x$ , the derivative of $π x$ with respect to $x$ is $π \frac{d}{d x} [x]$ .

$π \frac{d}{d x} [x]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$π \cdot 1$

Step 4.1.4

Multiply $π$ by $1$ .

$π$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π} \int \cos (u) \frac{1}{π} d u$

Step 5

Combine $\cos (u)$ and $\frac{1}{π}$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π} \int \frac{\cos (u)}{π} d u$

Step 6

Since $\frac{1}{π}$ is constant with respect to $u$ , move $\frac{1}{π}$ out of the integral.

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π} (\frac{1}{π} \int \cos (u) d u)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{π}$ by $\frac{1}{π}$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π π} \int \cos (u) d u$

Step 7.2

Raise $π$ to the power of $1$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π^{1} π} \int \cos (u) d u$

Step 7.3

Raise $π$ to the power of $1$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π^{1} π^{1}} \int \cos (u) d u$

Step 7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π^{1 + 1}} \int \cos (u) d u$

Step 7.5

Add $1$ and $1$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π^{2}} \int \cos (u) d u$

Step 8

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π^{2}} (\sin (u) + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $(x - 1) (- \frac{\cos (π x)}{π}) + \frac{1}{π^{2}} (\sin (u) + C)$ as $- x \frac{\cos (π x)}{π} + \frac{\cos (π x)}{π} + \frac{1}{π^{2}} \sin (u) + C$ .

$- x \frac{\cos (π x)}{π} + \frac{\cos (π x)}{π} + \frac{1}{π^{2}} \sin (u) + C$

Step 9.2

Combine $\frac{\cos (π x)}{π}$ and $x$ .

$- \frac{\cos (π x) x}{π} + \frac{\cos (π x)}{π} + \frac{1}{π^{2}} \sin (u) + C$

Step 10

Replace all occurrences of $u$ with $π x$ .

$- \frac{\cos (π x) x}{π} + \frac{\cos (π x)}{π} + \frac{1}{π^{2}} \sin (π x) + C$

Step 11

Simplify.

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Step 11.1

Reorder factors in $\frac{\cos (π x) x}{π}$ .

$- \frac{x \cos (π x)}{π} + \frac{\cos (π x)}{π} + \frac{1}{π^{2}} \sin (π x) + C$

Step 11.2

Reorder terms.

$- \frac{1}{π} x \cos (π x) + \frac{1}{π} \cos (π x) + \frac{1}{π^{2}} \sin (π x) + C$